My picture was deleted for some reason, but that picture was just an image I took off google the parameters in it were not the ones I was using. I was just using that image as an example since you didn't know what an offset crank mechanism is. However, you are correct in that there is an additional parameter which is the offset distance between the origin and the slider.
As for what I meant by "the displacement equation will stay the same":
In an offset model the displacement can be calculated the same way. The cos(theta of link 1) + the cos(fi of link 2) still equals to the displacement of the slider. Adding a vertical parameter will not change the horizontal parameter.
I am not calculating forces, but your last comment makes me think if the forces are different than the accelerations are different. Thus, I can conclude that I should derive new equations for my model(the offset crank mechanism)